not being split. Manolescu proved the latter by exploiting \(\mathrm {Pin}(2)\) -symmetry in Seiberg-Witten theory. Since then, related ideas in the context of Heegaard Floer homology [1]} have been used to show that \(\Theta _3\)  admits a \(\mathbb {Z}^{\infty }\)  summand [2]}. Despite this, still very little is known about the algebraic structure of \(\Theta _3\) : for example, it is an open question whether there are torsion or divisible elements in \(\Theta _3\) .
